Seams in the wires of paper machines are usually formed by reducing the thickness of the wire ends on the wear side facing away from the web, and the portion with the reduced thickness is folded double, whereafter a seam spiral is attached to the loops formed by the folded threads for interconnecting the wire ends. The folded wire end, the thickness of which is substantially equal to the original wire thickness when the thickness has been reduced appropriately, is stitched to form a substantially integral wire portion by means of thread stitches extending in the transverse direction of the wire. In prior art wire solutions, where the warp threads forming the surface extend in the direction of travel of the wire, that is, in its longitudinal direction, a problem is caused by the fact that the thread used for sewing the stitch is exposed to rubbing and wears out when the wire is used so that it gradually begins to break. As a result of this, the seam gradually begins to get weaker, and so the wire has to be replaced with a new one before the wire gauze itself needs to be replaced. This, in turn, involves extra, unnecessary costs.